Hazzazi, Mohammad Mazyad M (2019) On decompositions of finite projective planes and their applications. Doctoral thesis (PhD), University of Sussex.

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Abstract

Let PG(2; q) be the projective plane over the field Fq. Singer [19] notes that PG(2; q) has a cyclic group of order q2 + q + 1 that permutes the points of PG(2; q) in a single cycle. A k-arc set of k points no three of which are collinear. A k-arc is called complete if it is not contained in a (k + 1)-arc of PG(2; q).

By taking the orbits of points under a proper subgroup of a single cycle, one can decompose the projective plane PG(2; qk) into disjoint copies of subplanes isomorphic to PG(2; q) if and only if k is not divisible by three. Moreover, by taking the orbits of points under a proper subgroup, one can decompose the projective plane PG(2; q2) into disjoint copies of complete (q2 - q + 1)-arcs. In this thesis, our main problem is to classify (up to isomorphism) the different types of decompositions of PG(2 ;q2) for q = 3; 4; 5; 7, namely subplanes and arcs. We further illustrate some of the connections between these subgeometry decompositions and other areas of combinatorial interest; in particular, we explain the relationship between coding theory and projective spaces and describe the links with Hermitian unital. Furthermore, projective codes are obtained by taking the disjoint union of such subgeometries.

Item Type:	Thesis (Doctoral)
Schools and Departments:	School of Mathematical and Physical Sciences > Mathematics
Subjects:	Q Science > QA Mathematics > QA0440 Geometry. Trigonometry. Topology > QA0564 Algebraic geometry
Depositing User:	Library Cataloguing
Date Deposited:	07 Jun 2019 14:45
Last Modified:	07 Jun 2019 14:45
URI:	http://sro.sussex.ac.uk/id/eprint/83855

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